This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A258773 #35 Mar 01 2025 22:49:20
%S A258773 1,0,1,0,-2,4,0,3,-24,27,0,-4,96,-324,256,0,5,-320,2430,-5120,3125,0,
%T A258773 -6,960,-14580,61440,-93750,46656,0,7,-2688,76545,-573440,1640625,
%U A258773 -1959552,823543,0,-8,7168,-367416,4587520,-21875000,47029248,-46118408,16777216
%N A258773 Triangle read by rows, T(n,k) = (-1)^(n-k)*C(n,k)*k^n, for n >= 0 and 0 <= k <= n.
%C A258773 The row polynomials are p(0, x) = 1, and p(n, x) = Eu(x)^n (x-1)^n, for n >= 1, where Eu(x) := x*d/dx is the Euler derivative with respect to x. See A075513. - _Wolfdieter Lang_, Oct 12 2022
%C A258773 Coefficients of the Sidi polynomials (-1)^n*x*D_{n,0,n}(x), for n >= 0, where D_{k,n,m}(z) is given in Theorem 4.2., p. 862, of Sidi [1980]. - _Wolfdieter Lang_, Apr 10 2023
%H A258773 G. C. Greubel, <a href="/A258773/b258773.txt">Table of n, a(n) for n = 0..1326</a>
%H A258773 Avram Sidi, <a href="https://doi.org/10.1090/S0025-5718-1980-0572861-2">Numerical Quadrature and Nonlinear Sequence Transformations; Unified Rules for Efficient Computation of Integrals with Algebraic and Logarithmic Endpoint Singularities</a>, Math. Comp., 35 (1980), 851-874.
%F A258773 Sum_{k=0..n} T(n,k) = n!.
%F A258773 Sum_{k=0..n} |T(n,k)| = A072034(n).
%F A258773 Sum_{n>=0} Sum_{k=0..n} T(n,k) x^k y^n/n! = 1/(1 + W(-x*y*exp(-y))) where W is the Lambert W function. - _Robert Israel_, Dec 16 2015
%F A258773 T(n,n) = A000312(n). - _Peter Luschny_, Dec 17 2015
%F A258773 T(n, k+1) = n * A075513(n, k) if n > 0. - _Michael Somos_, May 13 2018
%e A258773 Triangle begins:
%e A258773 [1]
%e A258773 [0, 1]
%e A258773 [0, -2, 4]
%e A258773 [0, 3, -24, 27]
%e A258773 [0, -4, 96, -324, 256]
%e A258773 [0, 5, -320, 2430, -5120, 3125]
%e A258773 [0, -6, 960, -14580, 61440, -93750, 46656]
%e A258773 [0, 7, -2688, 76545, -573440, 1640625, -1959552, 823543]
%p A258773 seq(seq((-1)^(n-k)*binomial(n, k)*k^n, k=0..n), n=0..8);
%p A258773 T_row := proc(n) (-1)^n*(1-exp(x))^n/n!; diff(%,[x$n]); subs(exp(x)=t, n!*expand(%,x)); CoefficientList(%,t) end: seq(print(T_row(n)), n=0..7);
%t A258773 Flatten@Table[(-1)^(n - k) Binomial[n, k] k^n, {n, 0 , 10}, {k, 0, n}] (* _G. C. Greubel_, Dec 17 2015 *)
%Y A258773 Cf. A000142, A000312, A072034, A075513, A154715.
%K A258773 sign,easy,tabl
%O A258773 0,5
%A A258773 _Peter Luschny_, Jun 09 2015